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Chapter 1: Problem 45

The projection of a vector \(\vec{r}=3 \hat{i}+\hat{j}+2 \hat{k}\) on the \(x-y\) plane has magnitude (A) 3 (B) 4 (C) \(\sqrt{14}\) (D) \(\sqrt{10}\)

Short Answer

Expert verified
The magnitude of the projection of the vector \(\vec{r}=3 \hat{i}+\hat{j}+2 \hat{k}\) onto the \(x-y\) plane is (D) \(\sqrt{10}\).

Step by step solution

01

Finding the projection of the vector onto the x-y plane

To find the projection of a vector \(\vec{r}=3 \hat{i}+\hat{j}+2 \hat{k}\) onto the \(x-y\) plane, we need to eliminate the \(z\)-component. In this case, since the \(z\)-component is \(2 \hat{k}\), the projection of the vector onto the \(x-y\) plane will be \(\vec{p}=3 \hat{i}+\hat{j}\).
02

Finding the magnitude of the projected vector

Now, we need to find the magnitude of the projected vector, which is defined as the square root of the sum of the squares of the components. In this case, the magnitude of the projected vector in the \(x-y\) plane can be calculated as follows: \[ |\vec{p}| = \sqrt{(3 \hat{i})^2+(\hat{j})^2}= \sqrt{3^2+1^2}= \sqrt{10} \] So, the magnitude of the projection of the vector \(\vec{r}\) onto the \(x-y\) plane is \(\sqrt{10}\). The correct answer is (D) \(\sqrt{10}\).

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